This paper continues the study that began in [1,2] of the Cauchy problem for (x,t)RNR+ for three higher-order degenerate quasilinear partial differential equations (PDEs), as basic models, ut=(-1)m+1m(|u|nu)+| u|nu,utt=(-1)m+1m(| u|nu)+|u|nu,ut=(-1)m+1[ m(|u|nu)]x1+(| u|nu)x1, where n0 is a fixed exponent and m is the (m2)th iteration of the Laplacian. A diverse class of degenerate PDEs from various areas of applications of three types: parabolic, hyperbolic, and nonlinear dispersion, is dealt with. General local, global, and blow-up features of such PDEs are studied on the basis of their blow-up similarity or traveling wave (for the last one) solutions. In [1,2], the LusternikSchnirel'man category theory of variational calculus and fibering methods were applied. The case m=2 and n0 was studied in greater detail analytically and numerically. Here, more attention is paid to a combination of a Cartesian approximation and fibering to get new compactly supported similarity patterns. Using numerics, such compactly supported solutions are constructed for m=3 and for higher orders. The "smother" case of negative n0 is included, with a typical "fast diffusionabsorption" parabolic PDE: ut=(-1)m+1m(|u|nu)-| u|nu,where n(-1,0), which admits finite-time extinction rather than blow-up. Finally, a homotopy approach is developed for some kind of classification of various patterns obtained by variational and other methods. Using a variety of analytic, variational, qualitative, and numerical methods allows us to justify that the above PDEs admit an infinite countable set of countable families of compactly supported blow-up (extinction) or traveling wave solutions.
Galaktionov, V. A., Mitidieri, E., & Pohozaev, S. I. (2011). Variational approach to complicated similarity solutions of higher-order nonlinear PDEs. II. Nonlinear Analysis: Real World Applications, 12(4), 2435-2466. https://doi.org/10.1016/j.nonrwa.2011.03.001